Geometric Characteristics Of Lupas(?) -Bézier Curve

Computer Aided Geometric Design(CAGD)is an interdiscipline that grew up rapidly with the development of modern industries,such as the manufacturing of aircrafts,ships and automobiles.CAGD is mainly used for the modeling and research of free-form curves and surfaces.The classical B?ezier curves play a key role in CAGD.Lupas(?)-Bézier curves are generalized B?ezier curves involving -integer,and have many good properties,such as geometric invariance,affine invariance and -inverse symmetry.If = 1,Lupas(?)-Bézier curves reduce to the classical B?ezier curves.This paper studies the geometric characteristics of Lupas(?)-Bézier curves.The main results are as follows:Firstly,we study the relationships between quadratic Lupas(?)-Bézier curves and conic sections.Based on the shoulder point and the parametric midpoint,the classifications of conic sections that represented by quadratic Lupas(?)-Bézier curves are discussed,in which the position of shoulder point and the parametric midpoint show the tension and the distribution of the points on the curves,respectively.On the contrary,the position of shoulder point and parametric midpoint can obtain the shape parameter ,which determine the Lupas(?)-Bézier curves.If = 1,the shoulder point coincide with the parametric midpoint.By the construction of quadratic Lupas(?)-Bézier curves,we obtain that quadratic Lupas(?)-Bézier curves can represent all the parabolic arcs and hyperbolic arcs.By computation,we also obtain the geometric parameters of quadratic Lupas(?)-Bézier curves,such as focus,vertex,centre and so on.Secondly,we analyze the relationships of Lupas(?)-Bézier curves,rational B?ezier curves and weighted Lupas(?)-Bézier curves.The relationships of Lupas(?)-Bézier curves and rational B?ezier curves are derived based on the two basis functions.we also obtain the relationships of weighted Lupas(?)-Bézier curves and rational B?ezier curves.As well as,the relationships of two weighted Lupas(?)-Bézier curves with different shape parameters are discussed.Finally,a new de Casteljau algorithm and its applications in the subdivision of Lupas(?)-Bézier curves are given.Based on the discussion of shoulder point,a new de Casteljau algorithm,which evaluates the point on the curves as point of tangency,is derived.The algorithm can be applied to the subdivision of Lupas(?)-Bézier curves,and the shape parameters of sub-curves can be obtained from the formula.The subdivisions of quadratic Lupas(?)-Bézier curves at shoulder point and parametric midpoint are discussed and the speeds of convergence are compared,respectively.We obtain the subdivisions at shoulder point is more superior.In the end,the new de Casteljau algorithm is used to achieve a method of computing the approximate arc length of the quadratic Lupas(?)-Bézier curves.